Comparing Survival Distributions in the Presence of Dependent Censoring: Asymptotic Validity and Bias-corrections of the Logrank Test

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Abstract

We study the asymptotic properties of the logrank
and stratified logrank tests under different types of assumptions
regarding the dependence of the censoring and the survival times.
When the treatment group and the covariates are conditionally
independent given that the subject is still at risk, the logrank
statistic is asymptotically standard normally distributed under the
null hypothesis of no treatment effect. Under this assumption, the
stratified logrank statistic has asymptotic properties similar to
logrank statistic.
However, if the assumption of conditional independence of the
treatment and covariates given the at risk indicator fails, then the
logrank test statistic is generally biased and the bias generally
increases in proportional to the square root of the sample size. We
provide general formulas for the asymptotic bias and variance. We
also establish a contiguous alternative theory regarding small
violations of the assumption as well as of the usually considered
small differences between treatment and control group survival
hazards.
We discuss and extend an available bias-correction method of
DiRienzo and Lagakos (2001a), especially with respect to the
practical use of this method with unknown and estimated distribution
function for censoring given treatment group and covariates. We
obtain the correct asymptotic distribution of the bias-corrected
test statistic when stratumwise Kaplan-Meier estimators of the
conditional censoring distribution are substituted into it. Within
this framework, we prove the asymptotic unbiasedness of the
corrected test and find a consistent variance estimator.
Major theoretical results and motivations of future studies are
confirmed by a series of simulation studies.